Question

Sanford Corporation expects to have earnings this coming year of $3 per share. Sanford
plans to retain all of its earnings for the next two years. For the subsequent two years, the
firm will retain 50% of its earnings. It will then retain 20% of its earnings from that point
onward. Each year, retained earnings will be invested in new projects with an expected
return of 25% per year. Any earnings that are not retained will be paid out as dividends.
Assume Sanford’s share count remains constant and all earnings growth comes from the
investment of retained earnings. If Sanford’s equity cost of capital is 10%, what price
would you estimate for Sanford stock?
(a) What are the earnings and dividends over the first three years?

(b) At what rate will the earnings and dividends grow in year 4 and 5 s? What are
the dividends at the end of year 4 and 5?

(c) What price would you estimate for Sanford stock?

Answer 1

Solution

We have prepared a table below to show the change in EPS and the dividend payout per share based on the retention ratios provided in the problem.

1. Since share count remains constant, we assume that the existing investments would continue to earn the same Earning per share (EPS) of $3 each year. Hence the Earnings per share (Fixed) row shows a fixed EPS of $3.

2. All earnings growth comes from the investment of retained earnings. This means that only fresh investments made from retained earnings would contribute to increase in EPS. We are given that retained earnings invested in new projects have an expected return of 25% per year. Hence, we multiply 25% into the previous year's retention per share to get the row 'Earnings on reinvested retention'

3. Total earnings per share is the sum of the two rows above.

4. Dividend Payout (%) = 1 - Retention (%)

5. 'Retention per share' is the 'Retention ratio (%)' row multiplied with the 'Total Earnings per share' row

6. Dividend payout per share is the 'Dividend Payout (%)' row multiplied with the 'Total Earnings per share' row

	Years
	1	2	3	4	5	6	7	8
Earnings per share (Fixed)	3	3	3	3	3	3	3	3
Earnings on reinvested retention 25%		0.75	0.94	0.49	0.44	0.17	0.16	0.16
Total Earnings per share	3	3.75	3.94	3.49	3.44	3.17	3.16	3.16
Retention ratio (%) (given)	100%	100%	50%	50%	20%	20%	20%	20%
Dividend Payout (%) (1 -Retention %)	0%	0%	50%	50%	80%	80%	80%	80%
Retention per share	3	3.75	1.97	1.75	0.69	0.63	0.63	0.63
Dividend payout per share	0	0	1.97	1.75	2.75	2.54	2.53	2.53

a) Earnings over the first three years: $3, $3.75 and $3.94

Dividends over the first three years: $0, $0 and $1.97

b) Rate at which earnings grow in year 4 and 5:

Earnings

Year 4: (3.49-3.94) / 3.94 = -0.114213 or (-) 11.42%

Year 5: (3.44-3.49) / 3.49 = - 0.014326 or (-)1.43%

Dividend

Year 4: (1.75-1.97) / 1.97 = -0.111675 or (-)11.17%

Year 5 (2.75-1.75) / 1.75 = 0.571429 or 57.14%

Dividends at end of year 4 and 5 are $1.75 and $2.75 respectively

c) Price of Sanford stock = Discounted PV of all dividends at rate of 10% (cost of capital)

From year 7 we see dividend is constant at 2.53. So value of the dividends from year 7 at the end of year 6 by the Dividend Discount Model

_PV6 = D₇ / (k_e - g)

_PV6 = Present value at end of year 6

D₇ = Dividend at end of year 7

k_e = cost of capital, more specifically cost of equity

g = growth rate of dividend which is 0% since dividend stays constant from year 7 onwards

PV₆ = 2.53 / 0.10

PV₆ = 25.30

Now to find the value of the share we need to discound and add up all dividends and the above present value

P₀ = D₁/(1+r)¹ + D₂/(1+r)² + D₃/(1+r)³ + D₄/(1+r)⁴ + D₅/(1+r)⁵ + (D₆ + PV₆)/(1+r)⁶

P₀ = 0 + 0 + 1.97/(1+0.1)³+1.75/(1+0.1)⁴+2.75/(1+0.1)⁵+ (2.54+25.30)/(1+0.1)⁶

P₀ = 0 + 0 + 1.48 + 1.195274 +1.707534 + 15.714954

P₀ = 20.097762

Hence the estimated price of Sanford stock per share is $20.10.